# stacks/stacks-project

Use the new lemmas

 @@ -2637,7 +2637,7 @@ \section{The Tate map} Lb^*\NL_{Y/X} \longrightarrow \NL_{Y'/X'} $$is a quasi-isomorphism by More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-NL}. More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-NL-flat}. Thus we get a canonical isomorphism b^*\det(\NL_{Y/X}) \to \det(\NL_{Y'/X'}) which sends the canonical section \delta(\NL_{Y/X}) to \delta(\NL_{Y'/ X'}), see  @@ -3824,6 +3824,25 @@ \section{The naive cotangent complex} Special case of Modules, Lemma \ref{modules-lemma-exact-sequence-NL}. \end{proof} \begin{lemma} \label{lemma-base-change-NL} Consider a cartesian diagram of schemes$$ \xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y } $$The canonical map (g')^*\NL_{X/Y} \to \NL_{X'/Y'} induces an isomorphism on H^0 and a surjection on H^{-1}. \end{lemma} \begin{proof} Translated into algebra this is More on Algebra, Lemma \ref{more-algebra-lemma-base-change-NL}. To do the translation use Lemma \ref{lemma-NL-affine}. \end{proof} \begin{lemma} \label{lemma-flat-base-change-NL} Consider a cartesian diagram of schemes @@ -3843,7 +3862,7 @@ \section{The naive cotangent complex} \end{proof} \begin{lemma} \label{lemma-base-change-NL} \label{lemma-base-change-NL-flat} Consider a cartesian diagram of schemes$$ \xymatrix{ Divided Power Algebra, Proposition \ref{dpa-proposition-regular-ideal}. \end{proof} \begin{lemma} \label{lemma-lci-NL} Let $f : X \to Y$ be a local complete intersection homomorphism. Then the naive cotangent complex $\NL_{X/Y}$ is a perfect object of $D(\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$. \end{lemma} \begin{proof} Translated into algebra this is More on Algebra, Lemma \ref{more-algebra-lemma-lci-NL}. To do the translation use Lemmas \ref{lemma-affine-lci} and \ref{lemma-NL-affine} as well as Derived Categories of Schemes, Lemmas \ref{perfect-lemma-affine-compare-bounded}, \ref{perfect-lemma-tor-dimension-affine} and \ref{perfect-lemma-perfect-affine}. \end{proof} \begin{lemma} \label{lemma-perfect-NL-lci} Let $f : X \to Y$ be a perfect morphism of locally Noetherian schemes.